Let $X$ be a smooth projective variety over an algebraically closed field$mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample andglobally generated line bundle. By using characteristic $p$ methods, we showthat $K_X otimes L^{otimes dim X} otimes A$ is globally generated and $K_Xotimes L^{otimes (dim X+1)} otimes A$ is very ample, provided the linebundle $A$ is nef but not numerically trivial, which generalizes results ofY.T. Siu, K.E. Smith and D.S. Keeler. On complex projective varieties, byinvestigating Kawamata-Viehweg-Nadel type vanishing theorems for vectorbundles, we also obtain the globally generation for adjoint vector bundles. Inparticular, for a holomorphic submersion $f:Xlongrightarrow Y$ with $L$ ampleand globally generated, and $A$ nef but not numerically trivial, we prove theglobal generation of $ f_*(K_{X/Y})^{otimes s}otimes K_Y otimes L^{otimesdim Y} otimes A$, which generalizes a result of Koll'ar.
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机译：让$ x $是一个平滑的投影态度，通过代数封闭的字段$ mathbb {k} $，任意特征。假设$ l $是一个充足的Andglobally生成的线包。通过使用特征$ P $方法，我们为k_x otimes l ^ { otimes dim x} otimes a $是全局生成的，$ k_x otimes l ^ { otimes（ zh x + 1）} otimes一美元非常充足，只要换行机$ a $是nef，但不是数值微不足道，这概括了结果。 SiU，K.E.史密斯和D.S. Keeler。在复杂的投影品种上，通过Vievientigating Kawamata-Viehweg-Nadel类型为载体的定理，我们还获得了伴随着伴随矢量捆绑的全球生成。单独的，对于全身浸没$ f：x longrightarrow y $ with $ l $ ampleand全球生成，而$ a $ nef但不是数值微不足道，我们证明了$ f _ *（k_ {x / y}）^ { otimes s} otimesk_y otimes l ^ { otimes dim y} otimes a $，它概括了Koll 'AR的结果。
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